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Bigraded notions of formality and Aeppli-Bott-Chern-Massey products

Aleksandar Milivojevic, Jonas Stelzig

Abstract

We introduce and study notions of bigraded formality for the algebra of forms on a complex manifold, along with their relation to higher Aeppli-Bott-Chern-Massey products which extend the case of triple products studied by Angella-Tomassini. We show that these Aeppli-Bott-Chern-Massey products on complex manifolds pull back non-trivially to the blow-up along a complex submanifold, as long their degree is less than the real codimension of the submanifold.

Bigraded notions of formality and Aeppli-Bott-Chern-Massey products

Abstract

We introduce and study notions of bigraded formality for the algebra of forms on a complex manifold, along with their relation to higher Aeppli-Bott-Chern-Massey products which extend the case of triple products studied by Angella-Tomassini. We show that these Aeppli-Bott-Chern-Massey products on complex manifolds pull back non-trivially to the blow-up along a complex submanifold, as long their degree is less than the real codimension of the submanifold.
Paper Structure (7 sections, 9 theorems, 37 equations, 4 figures, 1 table)

This paper contains 7 sections, 9 theorems, 37 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Aeppli--Bott--Chern higher products
  4. Eilenberg--Moore spectral sequence
  5. Ad hoc Massey products
  6. Bigraded notions of formality
  7. Massey products under blow-ups

Key Result

Proposition 2.4

Let $f:A\to B$ be an bigraded quasi--isomorphism of bounded double complexes.

Figures (4)

  • Figure 1: The indices in the Schweitzer complex.
  • Figure 2: The double complex of the Kodaira--Thurston surface.
  • Figure 3: The left-invariant double complex for $SU(3)/\left( U(1) \times U(1) \right)$
  • Figure 4: Relations between various discussed notions. A dotted arrow means that the source property does not imply the target property, with a counterexample labelling the arrow. KT denotes the Kodaira--Thurston surface, ST the manifold constructed by Sferruzza--Tomassini, Hopf denotes the Hopf surface, and Stein denotes any Stein manifold with a non-vanishing Massey product NT78.

Theorems & Definitions (53)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Remark 2.5
  • Remark 2.6
  • Definition 3.1
  • Remark 3.2
  • Example 3.3
  • ...and 43 more